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Theorem 2rmorex 3282
Description: Double restricted quantification with "at most one," analogous to 2moex 2343. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
2rmorex  |-  ( E* x  e.  A  E. y  e.  B  ph  ->  A. y  e.  B  E* x  e.  A  ph )
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( y)

Proof of Theorem 2rmorex
StepHypRef Expression
1 nfcv 2591 . . 3  |-  F/_ y A
2 nfre1 2893 . . 3  |-  F/ y E. y  e.  B  ph
31, 2nfrmo 3011 . 2  |-  F/ y E* x  e.  A  E. y  e.  B  ph
4 rspe 2890 . . . . . 6  |-  ( ( y  e.  B  /\  ph )  ->  E. y  e.  B  ph )
54ex 435 . . . . 5  |-  ( y  e.  B  ->  ( ph  ->  E. y  e.  B  ph ) )
65ralrimivw 2847 . . . 4  |-  ( y  e.  B  ->  A. x  e.  A  ( ph  ->  E. y  e.  B  ph ) )
7 rmoim 3277 . . . 4  |-  ( A. x  e.  A  ( ph  ->  E. y  e.  B  ph )  ->  ( E* x  e.  A  E. y  e.  B  ph  ->  E* x  e.  A  ph ) )
86, 7syl 17 . . 3  |-  ( y  e.  B  ->  ( E* x  e.  A  E. y  e.  B  ph 
->  E* x  e.  A  ph ) )
98com12 32 . 2  |-  ( E* x  e.  A  E. y  e.  B  ph  ->  ( y  e.  B  ->  E* x  e.  A  ph ) )
103, 9ralrimi 2832 1  |-  ( E* x  e.  A  E. y  e.  B  ph  ->  A. y  e.  B  E* x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1870   A.wral 2782   E.wrex 2783   E*wrmo 2785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-eu 2270  df-mo 2271  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-rmo 2790
This theorem is referenced by:  2reu2  37998
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